Semi-classical limit of the massive Klein-Gordon-Maxwell system toward the relativistic Euler-Maxwell system via an adapted modulated energy method

TL;DR

This work analyzes the semi-classical limit $\varepsilon\to 0$ of the massive Klein-Gordon-Maxwell (mKGM) system in (3+1)-D Minkowski space and proves convergence of the momentum $\mathbf{J}^\varepsilon$, density $\rho^\varepsilon$, and Faraday tensor $F^\varepsilon$ to their relativistic Euler-Maxwell (REM) counterparts, i.e. $\mathbf{J}^\varepsilon \to \mathbf{U}\rho$, $\rho^\varepsilon \to \rho$, and $F^\varepsilon \to F$, with the REM dynamics governing the limit. The analysis hinges on a gauge-invariant modulated stress-energy method, combined with a compactness argument, to establish strong convergence of the density and to control the electromagnetic field and momentum. The REM system is shown to be locally well-posed, and its relation to the relativistic massive Vlasov-Maxwell equations is made precise via a monokinetic measure, providing a bridge between the hydrodynamic and kinetic descriptions. A formal WKB derivation is given in an appendix to illustrate the REM emergence from mKGM, complementing the rigorous semi-classical results.

Abstract

We show that the momentum, the density, and the electromagnetic field associated with the massive KleinGordon-Maxwell equations converge in the semi-classical limit towards their respective equivalents associated with the relativistic Euler-Maxwell equations. The proof relies on a modulated stress-energy method and a compactness argument. We also give a proof of the well-posedness of the relativistic Euler-Maxwell equations and show how this system, and so the semi-classical limit of Klein-Gordon-Maxwell, is related to the relativistic massive Vlasov-Maxwell equations.

Theorems & Definitions (71)

• Definition 1.1
• Definition 1.2
• Remark 1.1
• Theorem 2.1
• Lemma 2.1
• Remark 2.1
• Lemma 2.2
• Remark 2.2
• Definition 3.1
• Definition 3.2